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- Scott Ahlgren, Ken Ono, Ken Ono
- 2004

If p is prime, then let φp denote the Legendre symbol modulo p and let p be the trivial character modulo p. As usual, let n+1Fn(x)p := n+1Fn „ φp, φp, . . . , φp p, . . . , p | x « p be the Gaussian hypergeometric series over Fp. For n > 2 the non-trivial values of n+1Fn(x)p have been difficult to obtain. Here we take the first step by obtaining a simple… (More)

- Scott Ahlgren, Ken Ono
- Proceedings of the National Academy of Sciences…
- 2001

Eighty years ago, Ramanujan conjectured and proved some striking congruences for the partition function modulo powers of 5, 7, and 11. Until recently, only a handful of further such congruences were known. Here we report that such congruences are much more widespread than was previously known, and we describe the theoretical framework that appears to… (More)

In a recent paper [A-O], the author and K. Ono study the “Gaussian” hypergeometric series 4F3(1)p over the finite field Fp. They describe relationships between values of these series, Fourier coefficients of modular forms, and the arithmetic of a certain algebraic variety. These relationships, together with tools from p-adic analysis and some unexpected… (More)

- Scott Ahlgren, Ken Ono
- 2005

We investigate divisibility properties of the traces and Hecke traces of singular moduli. In particular we prove that, if p is prime, these traces satisfy many congruences modulo powers of p which are described in terms of the factorization of p in imaginary quadratic fields. We also study generalizations of Lehner’s classical congruences j(z)|Up ≡ 744 (mod… (More)

- Scott Ahlgren, Ken Ono, Ken Ono
- 2004

The Langlands program predicts that certain Calabi-Yau threefolds are modular in the sense that their L-series correspond to the Mellin transforms of weight 4 newforms. Here we prove that the L-function of the threefold given by P4 i=1(xi + x −1 i ) = 0 is η 4(2z)η4(4z), the unique normalized eigenform in S4(Γ0(8)).

- MODULO, Scott Ahlgren, Matthew R. Boylan

If F (z) is a newform of weight 2λ and D is a fundamental discriminant, then let L(F ⊗ χD, s) be the usual twisted L-series. We study the algebraic parts of the central critical values of these twisted L-series modulo primes `. We show that if there are two D (subject to some local conditions) for which the algebraic part of L(F ⊗ χD , λ) is not 0 (mod `),… (More)

- Scott Ahlgren, Ken Ono
- 2001

Therefore, there are 5 partitions of the number 4. But (as happens in number theory) the seemingly simple business of counting the ways to break a number into parts leads quickly to some difficult and beautiful problems. Partitions play important roles in such diverse areas of mathematics as combinatorics, Lie theory, representation theory, mathematical… (More)

Given a projective variety defined over a number field, a central problem in number theory is to write down all of its local zeta functions. The case of elliptic curves over Q has recently been settled, and here we study the problem for K3 surfaces, a natural two-dimensional analog of elliptic curves. This is in general a difficult problem, since at any… (More)

In his lost notebook, Ramanujan recorded a formula relating a “character analogue” of the Dedekind eta-function, the integral of a quotient of eta-functions, and the value of a Dirichlet Lfunction at s = 2. Here we derive an infinite family of formulas which includes Ramanujan’s original formula as a special case. Our results depend on a representation of… (More)

is a weakly holomorphic modular form of integral or half-integral weight w2 on the congruence subgroup Γ1(N). By a weakly holomorphic modular form we mean a function f(z) which is holomorphic on the upper half-plane, meromorphic at the cusps, and which transforms in the usual way under the action of Γ1(N) on the upper half-plane (see, for example, [13] for… (More)